Theorem isnumbasgrplem2 43549

Description: If the (to be thought of as disjoint, although the proof does not require this) union of a set and its Hartogs number supports a group structure (more generally, a cancellative magma), then the set must be numerable. (Contributed by Stefan O'Rear, 9-Jul-2015.)

Assertion

Ref	Expression
isnumbasgrplem2	⊢ ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) → 𝑆 ∈ dom card)

Proof of Theorem isnumbasgrplem2

Dummy variables 𝑎 𝑏 𝑐 𝑑 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		basfn 17174	. . 3 ⊢ Base Fn V
2		ssv 3939	. . 3 ⊢ Grp ⊆ V
3		fvelimab 6899	. . 3 ⊢ ((Base Fn V ∧ Grp ⊆ V) → ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) ↔ ∃𝑥 ∈ Grp (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))))
4	1, 2, 3	mp2an 698	. 2 ⊢ ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) ↔ ∃𝑥 ∈ Grp (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)))
5		harcl 9464	. . . . . 6 ⊢ (har‘𝑆) ∈ On
6		onenon 9864	. . . . . 6 ⊢ ((har‘𝑆) ∈ On → (har‘𝑆) ∈ dom card)
7	5, 6	ax-mp 5	. . . . 5 ⊢ (har‘𝑆) ∈ dom card
8		xpnum 9866	. . . . 5 ⊢ (((har‘𝑆) ∈ dom card ∧ (har‘𝑆) ∈ dom card) → ((har‘𝑆) × (har‘𝑆)) ∈ dom card)
9	7, 7, 8	mp2an 698	. . . 4 ⊢ ((har‘𝑆) × (har‘𝑆)) ∈ dom card
10		ssun1 4107	. . . . . . . 8 ⊢ 𝑆 ⊆ (𝑆 ∪ (har‘𝑆))
11		simpr 485	. . . . . . . 8 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)))
12	10, 11	sseqtrrid 3958	. . . . . . 7 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ⊆ (Base‘𝑥))
13		fvex 6840	. . . . . . . 8 ⊢ (Base‘𝑥) ∈ V
14	13	ssex 5249	. . . . . . 7 ⊢ (𝑆 ⊆ (Base‘𝑥) → 𝑆 ∈ V)
15	12, 14	syl 17	. . . . . 6 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ∈ V)
16	7	a1i 11	. . . . . 6 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → (har‘𝑆) ∈ dom card)
17		simp1l 1204	. . . . . . . 8 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → 𝑥 ∈ Grp)
18	12	3ad2ant1 1139	. . . . . . . . 9 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → 𝑆 ⊆ (Base‘𝑥))
19		simp2 1143	. . . . . . . . 9 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → 𝑎 ∈ 𝑆)
20	18, 19	sseldd 3916	. . . . . . . 8 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → 𝑎 ∈ (Base‘𝑥))
21		ssun2 4108	. . . . . . . . . . 11 ⊢ (har‘𝑆) ⊆ (𝑆 ∪ (har‘𝑆))
22	21, 11	sseqtrrid 3958	. . . . . . . . . 10 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → (har‘𝑆) ⊆ (Base‘𝑥))
23	22	3ad2ant1 1139	. . . . . . . . 9 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → (har‘𝑆) ⊆ (Base‘𝑥))
24		simp3 1144	. . . . . . . . 9 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → 𝑐 ∈ (har‘𝑆))
25	23, 24	sseldd 3916	. . . . . . . 8 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → 𝑐 ∈ (Base‘𝑥))
26		eqid 2739	. . . . . . . . 9 ⊢ (Base‘𝑥) = (Base‘𝑥)
27		eqid 2739	. . . . . . . . 9 ⊢ (+g‘𝑥) = (+g‘𝑥)
28	26, 27	grpcl 18908	. . . . . . . 8 ⊢ ((𝑥 ∈ Grp ∧ 𝑎 ∈ (Base‘𝑥) ∧ 𝑐 ∈ (Base‘𝑥)) → (𝑎(+g‘𝑥)𝑐) ∈ (Base‘𝑥))
29	17, 20, 25, 28	syl3anc 1379	. . . . . . 7 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → (𝑎(+g‘𝑥)𝑐) ∈ (Base‘𝑥))
30		simp1r 1205	. . . . . . 7 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)))
31	29, 30	eleqtrd 2841	. . . . . 6 ⊢ (((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆 ∧ 𝑐 ∈ (har‘𝑆)) → (𝑎(+g‘𝑥)𝑐) ∈ (𝑆 ∪ (har‘𝑆)))
32		simplll 780	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑥 ∈ Grp)
33	22	ad2antrr 732	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → (har‘𝑆) ⊆ (Base‘𝑥))
34		simprl 776	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑐 ∈ (har‘𝑆))
35	33, 34	sseldd 3916	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑐 ∈ (Base‘𝑥))
36		simprr 778	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑑 ∈ (har‘𝑆))
37	33, 36	sseldd 3916	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑑 ∈ (Base‘𝑥))
38	12	ad2antrr 732	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑆 ⊆ (Base‘𝑥))
39		simplr 774	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑎 ∈ 𝑆)
40	38, 39	sseldd 3916	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → 𝑎 ∈ (Base‘𝑥))
41	26, 27	grplcan 18967	. . . . . . 7 ⊢ ((𝑥 ∈ Grp ∧ (𝑐 ∈ (Base‘𝑥) ∧ 𝑑 ∈ (Base‘𝑥) ∧ 𝑎 ∈ (Base‘𝑥))) → ((𝑎(+g‘𝑥)𝑐) = (𝑎(+g‘𝑥)𝑑) ↔ 𝑐 = 𝑑))
42	32, 35, 37, 40, 41	syl13anc 1380	. . . . . 6 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑎 ∈ 𝑆) ∧ (𝑐 ∈ (har‘𝑆) ∧ 𝑑 ∈ (har‘𝑆))) → ((𝑎(+g‘𝑥)𝑐) = (𝑎(+g‘𝑥)𝑑) ↔ 𝑐 = 𝑑))
43		simplll 780	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑥 ∈ Grp)
44	12	ad2antrr 732	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑆 ⊆ (Base‘𝑥))
45		simprr 778	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑑 ∈ 𝑆)
46	44, 45	sseldd 3916	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑑 ∈ (Base‘𝑥))
47		simprl 776	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑎 ∈ 𝑆)
48	44, 47	sseldd 3916	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑎 ∈ (Base‘𝑥))
49	22	ad2antrr 732	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → (har‘𝑆) ⊆ (Base‘𝑥))
50		simplr 774	. . . . . . . 8 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑏 ∈ (har‘𝑆))
51	49, 50	sseldd 3916	. . . . . . 7 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → 𝑏 ∈ (Base‘𝑥))
52	26, 27	grprcan 18940	. . . . . . 7 ⊢ ((𝑥 ∈ Grp ∧ (𝑑 ∈ (Base‘𝑥) ∧ 𝑎 ∈ (Base‘𝑥) ∧ 𝑏 ∈ (Base‘𝑥))) → ((𝑑(+g‘𝑥)𝑏) = (𝑎(+g‘𝑥)𝑏) ↔ 𝑑 = 𝑎))
53	43, 46, 48, 51, 52	syl13anc 1380	. . . . . 6 ⊢ ((((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) ∧ 𝑏 ∈ (har‘𝑆)) ∧ (𝑎 ∈ 𝑆 ∧ 𝑑 ∈ 𝑆)) → ((𝑑(+g‘𝑥)𝑏) = (𝑎(+g‘𝑥)𝑏) ↔ 𝑑 = 𝑎))
54		harndom 9467	. . . . . . 7 ⊢ ¬ (har‘𝑆) ≼ 𝑆
55	54	a1i 11	. . . . . 6 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → ¬ (har‘𝑆) ≼ 𝑆)
56	15, 16, 16, 31, 42, 53, 55	unxpwdom3 43540	. . . . 5 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ≼* ((har‘𝑆) × (har‘𝑆)))
57		wdomnumr 9977	. . . . . 6 ⊢ (((har‘𝑆) × (har‘𝑆)) ∈ dom card → (𝑆 ≼* ((har‘𝑆) × (har‘𝑆)) ↔ 𝑆 ≼ ((har‘𝑆) × (har‘𝑆))))
58	9, 57	ax-mp 5	. . . . 5 ⊢ (𝑆 ≼* ((har‘𝑆) × (har‘𝑆)) ↔ 𝑆 ≼ ((har‘𝑆) × (har‘𝑆)))
59	56, 58	sylib 219	. . . 4 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ≼ ((har‘𝑆) × (har‘𝑆)))
60		numdom 9951	. . . 4 ⊢ ((((har‘𝑆) × (har‘𝑆)) ∈ dom card ∧ 𝑆 ≼ ((har‘𝑆) × (har‘𝑆))) → 𝑆 ∈ dom card)
61	9, 59, 60	sylancr 593	. . 3 ⊢ ((𝑥 ∈ Grp ∧ (Base‘𝑥) = (𝑆 ∪ (har‘𝑆))) → 𝑆 ∈ dom card)
62	61	rexlimiva 3132	. 2 ⊢ (∃𝑥 ∈ Grp (Base‘𝑥) = (𝑆 ∪ (har‘𝑆)) → 𝑆 ∈ dom card)
63	4, 62	sylbi 218	1 ⊢ ((𝑆 ∪ (har‘𝑆)) ∈ (Base “ Grp) → 𝑆 ∈ dom card)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119  ∃wrex 3063  Vcvv 3431   ∪ cun 3881   ⊆ wss 3883   class class class wbr 5072   × cxp 5616  dom cdm 5618   “ cima 5621  Oncon0 6310   Fn wfn 6480  ‘cfv 6485  (class class class)co 7356   ≼ cdom 8881  harchar 9461   ≼^* cwdom 9469  cardccrd 9850  Basecbs 17170  +_gcplusg 17211  Grpcgrp 18900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-rep 5199  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-1cn 11087  ax-addcl 11089

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-ral 3054  df-rex 3064  df-rmo 3344  df-reu 3345  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3903  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-int 4878  df-iun 4923  df-br 5073  df-opab 5135  df-mpt 5154  df-tr 5180  df-id 5513  df-eprel 5518  df-po 5526  df-so 5527  df-fr 5571  df-se 5572  df-we 5573  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-pred 6252  df-ord 6313  df-on 6314  df-lim 6315  df-suc 6316  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-isom 6494  df-riota 7313  df-ov 7359  df-oprab 7360  df-mpo 7361  df-om 7807  df-1st 7931  df-2nd 7932  df-frecs 8221  df-wrecs 8252  df-recs 8301  df-rdg 8339  df-1o 8395  df-oadd 8399  df-omul 8400  df-er 8633  df-map 8765  df-en 8884  df-dom 8885  df-sdom 8886  df-oi 9415  df-har 9462  df-wdom 9470  df-card 9854  df-acn 9857  df-nn 12166  df-slot 17143  df-ndx 17155  df-base 17171  df-0g 17395  df-mgm 18599  df-sgrp 18678  df-mnd 18694  df-grp 18903  df-minusg 18904

This theorem is referenced by:  isnumbasabl  43551  isnumbasgrp  43552